Higham, Nicholas J. and Knight, Philip A.
(1993)
*Finite precision behavior of stationary iteration for solving singular systems.*
Linear Algebra and its Applications, 192.
pp. 165-186.
ISSN 0024-3795

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## Abstract

A stationary iterative method for solving a singular system Ax=b converges for any starting vector if limi→∞Gi exists, where G is the iteration matrix, and the solution to which it converges depends on the starting vector. We examine the behavior of stationary iteration in finite precision arithmetic. A pertubation bound for singular systems is derived and used to define forward stability of a numerical method. A rounding error analysis enables us to deduce conditions under which a stationary iterative method is forward stable or backward stable. The component of the forward error in the null space of A can grow linearly with the number of iterations, but it is innocuous as long as the iteration converges reasonably quickly. As special cases, we show that when A is symmetric positive semidefinite the Richardson iteration with optimal parameter is forward stable, and if A also has unit diagonal and property A, then the Gauss-Seidel method is both forward and backward stable. Two numerical examples are given to illustrate the analysis.

Item Type: | Article |
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Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis |

Depositing User: | Ms Lucy van Russelt |

Date Deposited: | 04 Jul 2006 |

Last Modified: | 20 Oct 2017 14:12 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/360 |

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